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We classify all possible extensions of a valuation from a ground field $K$ to a rational function field in one or several variables over $K$. We determine which value groups and residue fields can appear, and we show how to construct extensions having these value groups and residue fields. In particular, we give several constructions of extensions whose corresponding value group and residue field extensions are not finitely generated. In the case of a rational function field $K(x)$ in one variable, we consider the relative algebraic closure of $K$ in the henselization of $K(x)$ with respect to the given extension, and we show that this can be any countably generated separable-algebraic extension of $K$. In the ``tame case'', we show how to determine this relative algebraic closure. Finally, we apply our methods to power series fields and the $p$-adics. [ABSTRACT FROM AUTHOR]

We generalize the main result of an earlier paper by the authors (\textit{Exponentiation in power series fields}, Proc. Amer. Math. Soc. \textbf{125} (1997), 3177--3183) concerning the convex embeddings of a chain $\Gamma$ in a lexicographic power $\Delta ^{\Gamma}$. For a fixed non-empty chain $\Delta$, we derive necessary and sufficient conditions for the existence of non-empty solutions $\Gamma$ to each of the lexicographic functorial equations \[(\Delta ^{\Gamma})^{\leq 0} \simeq \Gamma\>, (\Delta ^{\Gamma}) \simeq \Gamma \mbox{ and } (\Delta ^{\Gamma})^{<0} \simeq \Gamma\;.\] [ABSTRACT FROM AUTHOR]

Abstract. We describe the valuation theoretic properties of the Hardy fields associated to models of /FORMULA>, where T is the theory of a polynomially bounded o-minimal expansion of the reals and $\exp$ is the real exponential function. We deduce that /FORMULA> has levels with parameters and is exponentially bounded. We establish a maximality property of $H(\mathbb{R}_{\rm an, powers})$, the Hardy field of the expansion by the restricted analytic functions and power functions. [ABSTRACT FROM AUTHOR]

We prove that for no nontrivial ordered abelian group $G$ does the ordered power series field $\R((G))$ admit an exponential, i.e. an isomorphism between its ordered additive group and its ordered multiplicative group of positive elements, but that there is a non-surjective logarithm. For an arbitrary ordered field $k$, no exponential on $k((G))$ is compatible, that is, induces an exponential on $k$ through the residue map. This is proved by showing that certain functional equations for lexicographic powers of ordered sets are not solvable. [ABSTRACT FROM AUTHOR]